Theorem meredithOLD 1201

Description: Carew Meredith's sole axiom for propositional calculus. This amazing formula is thought to be the shortest possible single axiom for propositional calculus with inference rule ax-mp 7, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 4, ax-2 5, and ax-3 6. Then from it we derive the Lukasiewicz axioms luk-1 1215, luk-2 1216, and luk-3 1217. Using these we finally re-derive our axioms as ax1 1226, ax2 1227, and ax3 1228, thus proving the equivalence of all three systems. C. A. Meredith, "Single Axioms for the Systems (C,N), (C,O) and (A,N) of the Two-Valued Propositional Calculus," The Journal of Computing Systems vol. 1 (1953), pp. 155-164. Meredith claimed to be close to a proof that this axiom is the shortest possible, but the proof was apparently never completed.

An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic somewhat late in life after attending talks by Lukasiewicz and produced many remarkable results such as this axiom. From his obituary: "He did logic whenever time and opportunity presented themselves, and he did it on whatever materials came to hand: in a pub, his favored pint of porter within reach, he would use the inside of cigarette packs to write proofs for logical colleagues."

Assertion

Ref	Expression
meredithOLD	|- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((ta -> ph) -> (th -> ph)))

Proof of Theorem meredithOLD

Step	Hyp	Ref	Expression
1		ax-3 6	. . . . 5 |- ((-. ch -> -. (-. ch -> (-. ph -> -. th))) -> ((-. ch -> (-. ph -> -. th)) -> ch))
2		pm2.21 92	. . . . . . 7 |- (-. ph -> (ph -> ps))
3	2	imim1i 19	. . . . . 6 |- (((ph -> ps) -> (-. ch -> -. th)) -> (-. ph -> (-. ch -> -. th)))
4	3	com23 36	. . . . 5 |- (((ph -> ps) -> (-. ch -> -. th)) -> (-. ch -> (-. ph -> -. th)))
5	1, 4	syl5 20	. . . 4 |- ((-. ch -> -. (-. ch -> (-. ph -> -. th))) -> (((ph -> ps) -> (-. ch -> -. th)) -> ch))
6	5	imim1i 19	. . 3 |- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((-. ch -> -. (-. ch -> (-. ph -> -. th))) -> ta))
7	6	con3d 111	. 2 |- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> (-. ta -> -. (-. ch -> -. (-. ch -> (-. ph -> -. th)))))
8		pm2.27 76	. . . . . . . . 9 |- (-. ch -> ((-. ch -> (-. ph -> -. th)) -> (-. ph -> -. th)))
9	8	impi 160	. . . . . . . 8 |- (-. (-. ch -> -. (-. ch -> (-. ph -> -. th))) -> (-. ph -> -. th))
10	9	com12 14	. . . . . . 7 |- (-. ph -> (-. (-. ch -> -. (-. ch -> (-. ph -> -. th))) -> -. th))
11	10	imim2d 28	. . . . . 6 |- (-. ph -> ((-. ta -> -. (-. ch -> -. (-. ch -> (-. ph -> -. th)))) -> (-. ta -> -. th)))
12	11	com12 14	. . . . 5 |- ((-. ta -> -. (-. ch -> -. (-. ch -> (-. ph -> -. th)))) -> (-. ph -> (-. ta -> -. th)))
13	12	a2d 16	. . . 4 |- ((-. ta -> -. (-. ch -> -. (-. ch -> (-. ph -> -. th)))) -> ((-. ph -> -. ta) -> (-. ph -> -. th)))
14		con3 110	. . . 4 |- ((ta -> ph) -> (-. ph -> -. ta))
15	13, 14	syl5 20	. . 3 |- ((-. ta -> -. (-. ch -> -. (-. ch -> (-. ph -> -. th)))) -> ((ta -> ph) -> (-. ph -> -. th)))
16		ax-3 6	. . 3 |- ((-. ph -> -. th) -> (th -> ph))
17	15, 16	syl6 25	. 2 |- ((-. ta -> -. (-. ch -> -. (-. ch -> (-. ph -> -. th)))) -> ((ta -> ph) -> (th -> ph)))
18	7, 17	syl 12	1 |- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((ta -> ph) -> (th -> ph)))

Syntax hints:  -. wn 2   -> wi 3

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

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